Properties of Roots and Radical Expressions

Radical expressions are expressions that contain roots, and a few properties of roots let you simplify them to a standard form. In brief: √(ab) = √a · √b and √(a/b) = √a / √b for nonnegative a and b. For example, √72 = √(36 · 2) = 6√2.

Simplifying radicals makes answers shorter, easier to compare and easier to add. The main idea is to pull out perfect powers from under the root sign. This guide lists the properties of roots, explains how to simplify, add and multiply radical expressions and shows common mistakes.

What are the properties of roots?

For nonnegative a and b (and positive index n), the following properties hold. They follow from the exponent laws applied to the form a^1/n.

• Product: ⁿ√(ab) = ⁿ√a · ⁿ√b
• Quotient: ⁿ√(a/b) = ⁿ√a / ⁿ√b, for b ≠ 0
• Power: (ⁿ√a)^m = ⁿ√(a^m)
• Root of a root: ^m√(ⁿ√a) = ^mn√a
• Cancelling: (ⁿ√a)ⁿ = a

The indices must match for the product and quotient properties. For example, √2 · ∛2 cannot be combined by multiplying the radicands. Using exponents, though, it equals 2^1/2 · 2^1/3 = 2^5/6.

How do you simplify a radical step by step?

A square root is in simplest form when the radicand has no perfect-square factor other than 1, contains no fractions and there is no root in a denominator.

1. Factor the radicand and find the largest perfect power that divides it.
2. Split the root with the product property.
3. Take the root of the perfect power and leave the rest under the radical.

Example 1: √72

72 = 36 · 2 and 36 is a perfect square. So √72 = √36 · √2 = 6√2.

Example 2: ∛54

54 = 27 · 2 and 27 is a perfect cube. So ∛54 = ∛27 · ∛2 = 3∛2.

Example 3: variables

√(18x⁵) = √(9 · 2 · x⁴ · x) = 3x²√(2x), for x ≥ 0.

How do you add, subtract and multiply radicals?

Radicals can be added or subtracted only if they are like radicals, meaning they have the same index and the same radicand. Then you combine coefficients, just like like terms: 3√5 + 4√5 = 7√5.

Sometimes radicals look different but become alike after simplifying. For √12 + √27, write √12 = 2√3 and √27 = 3√3. The sum is 5√3.

For multiplication, multiply coefficients and radicands separately: 2√3 · 5√6 = 10√18 = 10 · 3√2 = 30√2. For two binomials with radicals, use the distributive law: (√5 + 2)(√5 − 2) = 5 − 4 = 1. This difference of squares pattern is the key to rationalising denominators.

Simplified square roots table for radicands 8 to 50

The table lists every radicand from 8 to 50 that is not a perfect square but has a perfect-square factor. Radicands not listed, such as 2, 3, 5, 6 and 7, are already simplest.

Where do radical rules come from?

Greek mathematicians, and Euclid in particular, studied quantities that behave like square roots in geometric form. Book X of the Elements classifies irrational lines in great detail. Algebraic rules for radicals, as we write them today, developed in the Islamic world, in India and later in Europe, and they were put into modern notation in the sixteenth and seventeenth centuries.

Where are radical expressions used today?

Exact answers in geometry often contain radicals: the diagonal of a unit square is √2, and the height of an equilateral triangle with side 2 is √3. Keeping an answer in radical form is more precise than a rounded decimal. Engineers use simplified radicals when working with impedance and vector lengths, and the formulas for standard deviation and for solving quadratic equations produce them as well.

Which mistakes are common?

Adding unlike radicals

√2 + √3 is not √5. Without a common radicand the expression cannot be combined.

Stopping before the largest factor

√72 = 2√18 is correct but not fully simplified. Keep going until the radicand has no perfect-square factor: √72 = 6√2.

Applying the product rule to sums

√(a + b) is not √a + √b. For a = 9 and b = 16 the left side is 5 and the right side is 7.

Ignoring signs of variables

√(x²) = |x|. Writing x is only correct when x ≥ 0.